The Lateral Torsional Buckling of I Beams with Cross Beams

Abstract

High flexural-rigid cross beams at the joints of a main beam can function as restraining members to prevent rotation during lateral torsional buckling (LTB). With less flexural-rigid members, joint rotation results in a decrease of the main beam’s LTB critical moment. This paper elucidates the cross beam’s minimum flexural rigidity to prevent the main beam’s joint rotation during torsional buckling. It is assumed that material behaves elastically, the beam’s web doesn’t undergo distortions, and shear forces effects are neglected. The cross-beam’s flexural rigidity is represented by a spiral spring. Under buckling, this spring produces a torque moment, proportional to the joint rotation. The torque will disturb the main beam’s LTB equation system. By adjusting the spring constant, the joint rotation is minimized, thus reducing the torque’s disturbing effect within the equation. By neglecting this effect, the LTB equations at all fields of main beam are identical to the general buckling equations for constant moments. For n cross beams, 4 (n+1) integration constants are resulted. By applying the boundary conditions at beam ends and utilizing the geometry and natural boundary conditions at the joints, 4 (n+1) homogeneous equations for the integration constant are obtained. By conducting the trial and error method, the critical moment resulting in a zero determinant for the homogeneous equation coefficient matrix is acquired. Then, the LTB first-mode deformation shape can be drawn. The analysis shows that to achieve the critical moment, a main beam having a cross beam located at mid span needs the most optimum (smallest) flexural rigidity of cross beam than when it is in other locations (L/2). Observing the first-mode shape, for a certain spring constant value, the rotation at the joint will approach zero.